Moment problem

Example: Given the mean and variance ${\displaystyle \sigma ^{2}}$ (as well as all further cumulants equal 0) the normal distribution is the distribution solving the moment problem.

In mathematics, a moment problem arises as the result of trying to invert the mapping that takes a measure ${\displaystyle \mu }$ to the sequence of moments

${\displaystyle m_{n}=\int _{-\infty }^{\infty }x^{n}\,d\mu (x)\,.}$

More generally, one may consider

${\displaystyle m_{n}=\int _{-\infty }^{\infty }M_{n}(x)\,d\mu (x)\,.}$

for an arbitrary sequence of functions ${\displaystyle M_{n}}$.

Introduction

In the classical setting, ${\displaystyle \mu }$ is a measure on the real line, and ${\displaystyle M}$ is the sequence ${\displaystyle \{x^{n}:n=1,2,\dotsc \}}$. In this form the question appears in probability theory, asking whether there is a probability measure having specified mean, variance and so on, and whether it is unique.

There are three named classical moment problems: the Hamburger moment problem in which the support of ${\displaystyle \mu }$ is allowed to be the whole real line; the Stieltjes moment problem, for ${\displaystyle [0,\infty )}$; and the Hausdorff moment problem for a bounded interval, which without loss of generality may be taken as ${\displaystyle [0,1]}$.

The moment problem also extends to complex analysis as the trigonometric moment problem in which the Hankel matrices are replaced by Toeplitz matrices and the support of μ is the complex unit circle instead of the real line. ^[1]

Existence

A sequence of numbers ${\displaystyle m_{n}}$ is the sequence of moments of a measure ${\displaystyle \mu }$ if and only if a certain positivity condition is fulfilled; namely, the Hankel matrices ${\displaystyle H_{n}}$,

${\displaystyle (H_{n})_{ij}=m_{i+j}\,,}$

should be positive semi-definite. This is because a positive-semidefinite Hankel matrix corresponds to a linear functional ${\displaystyle \Lambda }$ such that ${\displaystyle \Lambda (x^{n})=m_{n}}$ and ${\displaystyle \Lambda (f^{2})\geq 0}$ (non-negative for sum of squares of polynomials). Assume ${\displaystyle \Lambda }$ can be extended to ${\displaystyle \mathbb {R} [x]^{*}}$. In the univariate case, a non-negative polynomial can always be written as a sum of squares. So the linear functional ${\displaystyle \Lambda }$ is positive for all the non-negative polynomials in the univariate case. By Haviland's theorem, the linear functional has a measure form, that is ${\displaystyle \Lambda (x^{n})=\int _{-\infty }^{\infty }x^{n}d\mu }$. A condition of similar form is necessary and sufficient for the existence of a measure ${\displaystyle \mu }$ supported on a given interval ${\displaystyle [a,b]}$.

One way to prove these results is to consider the linear functional ${\displaystyle \varphi }$ that sends a polynomial

${\displaystyle P(x)=\sum _{k}a_{k}x^{k}}$

to

${\displaystyle \sum _{k}a_{k}m_{k}.}$

If ${\displaystyle m_{k}}$ are the moments of some measure ${\displaystyle \mu }$ supported on ${\displaystyle [a,b]}$, then evidently

${\displaystyle \varphi (P)\geq 0}$ for any polynomial ${\displaystyle P}$ that is non-negative on ${\displaystyle [a,b]}$.

(1)

Vice versa, if ( 1 ) holds, one can apply the M. Riesz extension theorem and extend ${\displaystyle \varphi }$ to a functional on the space of continuous functions with compact support ${\displaystyle C_{c}([a,b])}$), so that

${\displaystyle \varphi (f)\geq 0}$ for any ${\displaystyle f\in C_{c}([a,b]),\;f\geq 0.}$

(2)

By the Riesz representation theorem, ( 2 ) holds iff there exists a measure ${\displaystyle \mu }$ supported on ${\displaystyle [a,b]}$, such that

${\displaystyle \varphi (f)=\int f\,d\mu }$

for every ${\displaystyle f\in C_{c}([a,b])}$.

Thus the existence of the measure ${\displaystyle \mu }$ is equivalent to ( 1 ). Using a representation theorem for positive polynomials on ${\displaystyle [a,b]}$, one can reformulate ( 1 ) as a condition on Hankel matrices. ^{[2] [3]}

Uniqueness (or determinacy)

See also: Carleman's condition and Krein's condition

The uniqueness of ${\displaystyle \mu }$ in the Hausdorff moment problem follows from the Weierstrass approximation theorem, which states that polynomials are dense under the uniform norm in the space of continuous functions on ${\displaystyle [0,1]}$. For the problem on an infinite interval, uniqueness is a more delicate question. ^[4] There are distributions, such as log-normal distributions, which have finite moments for all the positive integers but where other distributions have the same moments.

Formal solution

When the solution exists, it can be formally written using derivatives of the Dirac delta function as

${\displaystyle d\mu (x)=\rho (x)dx,\quad \rho (x)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n!}}\delta ^{(n)}(x)m_{n}}$.

The expression can be derived by taking the inverse Fourier transform of its characteristic function.

Variations

See also: Chebyshev–Markov–Stieltjes inequalities

An important variation is the truncated moment problem, which studies the properties of measures with fixed first k moments (for a finite k). Results on the truncated moment problem have numerous applications to extremal problems, optimisation and limit theorems in probability theory. ^[3]

Probability

The moment problem has applications to probability theory. The following is commonly used: ^[5]

Theorem (Fréchet-Shohat) — If ${\textstyle \mu }$ is a determinate measure (i.e. its moments determine it uniquely), and the measures ${\textstyle \mu _{n}}$ are such that $${\displaystyle \forall k\geq 0\quad \lim _{n\rightarrow \infty }m_{k}\left[\mu _{n}\right]=m_{k}[\mu ],}$$ then ${\textstyle \mu _{n}\rightarrow \mu }$ in distribution.

By checking Carleman's condition, we know that the standard normal distribution is a determinate measure, thus we have the following form of the central limit theorem:

Corollary — If a sequence of probability distributions ${\textstyle \nu _{n}}$ satisfy $${\displaystyle m_{2k}[\nu _{n}]\to {\frac {(2k)!}{2^{k}k!}};\quad m_{2k+1}[\nu _{n}]\to 0}$$ then ${\textstyle \nu _{n}}$ converges to ${\textstyle N(0,1)}$ in distribution.

See also

• Carleman's condition
• Hamburger moment problem
• Hankel matrix
• Hausdorff moment problem
• Moment (mathematics)
• Stieltjes moment problem
• Trigonometric moment problem

Notes

1. Schmüdgen 2017, p. 257.
2. Shohat & Tamarkin 1943.
3. Kreĭn & Nudel′man 1977.
4. Akhiezer 1965.
5. Sodin, Sasha (March 5, 2019). "The classical moment problem" (PDF). Archived (PDF) from the original on 1 Jul 2022.

References

• Shohat, James Alexander; Tamarkin, Jacob D. (1943). The Problem of Moments. New York: American mathematical society. ISBN   978-1-4704-1228-9.
• Akhiezer, Naum I. (1965). The classical moment problem and some related questions in analysis . New York: Hafner Publishing Co. (translated from the Russian by N. Kemmer)
• Kreĭn, M. G.; Nudel′man, A. A. (1977). The Markov Moment Problem and Extremal Problems. Translations of Mathematical Monographs. Providence, Rhode Island: American Mathematical Society. doi:10.1090/mmono/050. ISBN   978-0-8218-4500-4. ISSN   0065-9282.
• Schmüdgen, Konrad (2017). The Moment Problem. Graduate Texts in Mathematics. Vol. 277. Cham: Springer International Publishing. doi:10.1007/978-3-319-64546-9. ISBN   978-3-319-64545-2. ISSN   0072-5285.